Q. What is the y-intercept of the line given by the equation 2x + 5y - 10 = 0? (2019)
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Solution
Rearranging to y = (-2/5)x + 2, the y-intercept is 2.
Correct Answer:
A
— 2
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Q. What is the y-intercept of the line given by the equation 5x - 2y = 10? (2023)
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Solution
Rearranging to slope-intercept form: -2y = -5x + 10, thus y = (5/2)x - 5. The y-intercept is -5.
Correct Answer:
D
— -2
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Q. What is the y-intercept of the line represented by the equation 4x - y + 8 = 0? (2019)
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Solution
Setting x = 0 in the equation gives -y + 8 = 0, thus y = 8.
Correct Answer:
A
— 8
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Q. What type of matrix has all its diagonal elements as 1 and all other elements as 0? (2023)
A.
Identity matrix
B.
Zero matrix
C.
Diagonal matrix
D.
Scalar matrix
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Solution
An identity matrix is defined as a square matrix with all diagonal elements equal to 1 and all other elements equal to 0.
Correct Answer:
A
— Identity matrix
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Q. What type of matrix has all its elements equal to zero? (2022)
A.
Identity matrix
B.
Zero matrix
C.
Diagonal matrix
D.
Scalar matrix
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Solution
A matrix where all elements are zero is called a zero matrix.
Correct Answer:
B
— Zero matrix
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Q. What type of matrix has all its elements equal? (2022)
A.
Identity matrix
B.
Zero matrix
C.
Scalar matrix
D.
Diagonal matrix
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Solution
A scalar matrix is a square matrix in which all the elements are equal to the same scalar value.
Correct Answer:
C
— Scalar matrix
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Q. What type of matrix has the property A = -A? (2022)
A.
Symmetric matrix
B.
Skew-symmetric matrix
C.
Identity matrix
D.
Diagonal matrix
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Solution
A skew-symmetric matrix is defined by the property A = -A.
Correct Answer:
B
— Skew-symmetric matrix
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Q. Which identity represents the difference of squares?
A.
(a + b)² = a² + 2ab + b²
B.
(a - b)² = a² - 2ab + b²
C.
a² - b² = (a + b)(a - b)
D.
a² + b² = (a + b)² - 2ab
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Solution
The difference of squares is given by a² - b² = (a + b)(a - b).
Correct Answer:
C
— a² - b² = (a + b)(a - b)
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Q. Which identity represents the expansion of (a + b + c)²?
A.
a² + b² + c² + 2ab + 2bc + 2ca
B.
a² + b² + c² + 3abc
C.
a² + b² + c²
D.
2a² + 2b² + 2c²
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Solution
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.
Correct Answer:
A
— a² + b² + c² + 2ab + 2bc + 2ca
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Q. Which of the following complex numbers lies on the unit circle?
A.
1 + 0i
B.
0 + 1i
C.
1 + 1i
D.
0 + 0i
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Solution
A complex number z lies on the unit circle if |z| = 1. For z = 1 + 0i, |z| = √(1² + 0²) = 1. For z = 0 + 1i, |z| = √(0² + 1²) = 1. Both 1 + 0i and 0 + 1i lie on the unit circle.
Correct Answer:
A
— 1 + 0i
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Q. Which of the following data sets has the highest variance? A: {1, 2, 3}, B: {1, 1, 1}, C: {1, 5, 9}, D: {2, 2, 2}. (2020)
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Solution
Variance of A = 0.67, B = 0, C = 12, D = 0. Thus, C has the highest variance.
Correct Answer:
C
— C
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Q. Which of the following functions is continuous at all points?
A.
f(x) = 1/x
B.
f(x) = x^3
C.
f(x) = sqrt(x)
D.
f(x) = tan(x)
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Solution
f(x) = x^3 is a polynomial function, which is continuous everywhere.
Correct Answer:
B
— f(x) = x^3
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Q. Which of the following functions is continuous at x = 0?
A.
f(x) = 1/x
B.
f(x) = e^x
C.
f(x) = tan(x)
D.
f(x) = 1/(x^2 + 1)
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Solution
The function f(x) = e^x is continuous everywhere, including at x = 0.
Correct Answer:
B
— f(x) = e^x
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Q. Which of the following functions is continuous on the interval [0, 1]?
A.
f(x) = 1/x
B.
f(x) = x^3
C.
f(x) = sqrt(x)
D.
f(x) = 1/(x-1)
Show solution
Solution
f(x) = x^3 is a polynomial function and is continuous on the interval [0, 1].
Correct Answer:
B
— f(x) = x^3
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Q. Which of the following is a characteristic of a diagonal matrix? (2022)
A.
All elements are zero
B.
Only diagonal elements are non-zero
C.
All elements are equal
D.
It is always square
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Solution
A diagonal matrix has non-zero elements only on its diagonal, while all off-diagonal elements are zero.
Correct Answer:
B
— Only diagonal elements are non-zero
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Q. Which of the following is a property of an identity matrix? (2019)
A.
It is always square
B.
It has all elements as 1
C.
It is always a zero matrix
D.
It has a determinant of 0
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Solution
An identity matrix is always square and has 1s on the diagonal and 0s elsewhere.
Correct Answer:
A
— It is always square
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Q. Which of the following is a singular matrix? (2019)
A.
[[1, 2], [3, 4]]
B.
[[2, 3], [4, 6]]
C.
[[5, 0], [0, 5]]
D.
[[0, 1], [1, 0]]
Show solution
Solution
A singular matrix is one that does not have an inverse, which occurs when its determinant is zero. The matrix [[2, 3], [4, 6]] is singular.
Correct Answer:
B
— [[2, 3], [4, 6]]
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Q. Which of the following is a square matrix? (2020)
A.
2x3
B.
3x3
C.
4x2
D.
5x1
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Solution
A square matrix is one where the number of rows is equal to the number of columns. Among the options, only 3x3 has equal rows and columns.
Correct Answer:
B
— 3x3
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Q. Which of the following is a symmetric matrix? (2020)
A.
[[1, 2], [3, 4]]
B.
[[1, 2], [2, 1]]
C.
[[1, 0], [0, 1]]
D.
[[0, 1], [1, 0]]
Show solution
Solution
A symmetric matrix is one that is equal to its transpose. The matrix [[1, 2], [2, 1]] is symmetric.
Correct Answer:
B
— [[1, 2], [2, 1]]
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Q. Which of the following is an identity matrix? (2019)
A.
[[1, 0], [0, 1]]
B.
[[1, 1], [1, 1]]
C.
[[0, 1], [1, 0]]
D.
[[2, 0], [0, 2]]
Show solution
Solution
An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. The matrix [[1, 0], [0, 1]] fits this definition.
Correct Answer:
A
— [[1, 0], [0, 1]]
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Q. Which of the following is an irrational number?
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Solution
√4 = 2, 0.25 = 1/4, and 1 is rational. √2 is irrational.
Correct Answer:
B
— √2
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Q. Which of the following is equal to 0.125?
A.
1/8
B.
1/4
C.
1/2
D.
1/16
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Solution
0.125 is equal to 1/8.
Correct Answer:
A
— 1/8
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Q. Which of the following is equal to 0.25? (2023)
A.
1/2
B.
1/4
C.
1/3
D.
1/5
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Solution
0.25 is equal to 1/4.
Correct Answer:
B
— 1/4
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Q. Which of the following is equal to log_10(0.01)? (2019)
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Solution
log_10(0.01) = log_10(10^-2) = -2.
Correct Answer:
B
— -2
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Q. Which of the following is equivalent to log_2(32)?
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Solution
log_2(32) = log_2(2^5) = 5.
Correct Answer:
B
— 5
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Q. Which of the following is not a rational number? (2023)
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Solution
√3 is an irrational number, while 1/2, 0, and 3 are rational.
Correct Answer:
C
— √3
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Q. Which of the following is the conjugate of the complex number z = 2 - 3i?
A.
2 + 3i
B.
2 - 3i
C.
3 - 2i
D.
3 + 2i
Show solution
Solution
The conjugate of a complex number z = a + bi is given by z* = a - bi. Here, the conjugate is 2 + 3i.
Correct Answer:
A
— 2 + 3i
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Q. Which of the following is the conjugate of the complex number z = 5 - 2i? (2021)
A.
5 + 2i
B.
5 - 2i
C.
-5 + 2i
D.
-5 - 2i
Show solution
Solution
The conjugate of z = 5 - 2i is z* = 5 + 2i.
Correct Answer:
A
— 5 + 2i
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Q. Which of the following is the conjugate of the complex number z = 5 - 3i?
A.
5 + 3i
B.
5 - 3i
C.
3 - 5i
D.
3 + 5i
Show solution
Solution
The conjugate of a complex number z = a + bi is given by z* = a - bi. Here, the conjugate of 5 - 3i is 5 + 3i.
Correct Answer:
A
— 5 + 3i
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Q. Which of the following is the correct expansion of (x - 3)²?
A.
x² - 6x + 9
B.
x² + 6x + 9
C.
x² - 9
D.
x² - 3
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Solution
(x - 3)² = x² - 2(3)x + 3² = x² - 6x + 9.
Correct Answer:
A
— x² - 6x + 9
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